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Excel Mastermind with Optimal Strategy

MASTERMIND STRATEGIES

In the classical Mastermind game where the secret code is a 4 digit repeatable combination of 6 different colours, there are in fact just 5 unique types of first guesses to the game. And a player's guess will be scored according to its correct (colour and position) and semi-correct (correct colour but wrong position) digits. If we multiply correct number of digits by 10 and add to that number semi-correct digits in the guess, we can show each score by a 2-digit number: 00, 01, 02, ......, 30, and 40. Note that 31 is not a valid score. The table below shows the number of remaining valid possibilities after the first guess for each of the possible combinations and for each possible score. You can see which game strategy picks which first guess and why it does so by looking at the first column representing all the possible first guesses for each classical Mastermind strategy:

1111 1112 1122 1123 1234
00 625 256 256 81 16
01 0 308 256 276 152
02 0 61 96 222 312
03 0 0 16 44 136
04 0 0 1 2 9
10 500 317 256 182 108
11 0 156 208 230 252
12 0 27 36 84 132
13 0 0 0 4 8
20 150 123 114 105 96
21 0 24 32 40 48
22 0 3 4 5 6
30 20 20 20 20 20
40 1 1 1 1 1
Total: 1296 1296 1296 1296 1296
Simple 1 2 8 9 52
Worst Case 625 317 256 276 312
Entropy 1.5 2.7 2.9 3.0 3.1
Expected Size 512.0 235.9 204.5 185.3 188.2
Most Parts 5 11 13 14 14

Game Solution Tree Averages

This table below shows the game solution tree averages for all the classical strategies as well as the Optimal Strategy:

Strategy Total Guesses Average Maximum Guesses 6 or more Guesses Author Year
Optimal (Max: 6) 5625 4.34028 6 1 Lai 1993
Optimal (Max: 5) 5626 4.34105 5 0 Lai 1993
Weighted Entropy 5646 4.35648 6 3 Gur 2021
Most parts 5668 4.37346 6 7 Kooi 2005
Expected Size 5696 4.39506 6 3 Irving 1979
Entropy 5722 4.41512 6 12 Neuwirth 1982
Worst Case (Min-Max) 5801 4.47608 5 0 Knuth 1977
Simple 7471 5.76466 9 853 Shapiro 1983

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